7th Grade Chapter 5
Chapter 5 - Operations with Fractions
5-1 Comparing and Ordering Rational Numbers
A good method for comparing two fractions is to find a common denominator. The least common denominator (LCD) is the least common multiple (LCM) of the denominators. You can list the multiples of each number until you find a common multiple. Or, you can write the prime factorization of each number and use the greatest power of each prime factor. We use the phrase “take the left overs” after highlighting the common factors in the numerator and denominator. Listing the multiples may be easier for lesser numbers whereas prime factorization may be best for greater ones and numbers that include variables.
5-2 Fractions and Decimals
A rational number is any number you can write as a ratio a/b of two integers a and b, where b does not equal 0. Both terminating and repeating decimals can be written as the ratio of two integers. Decimals that are neither terminating nor repeating are irrational numbers and cannot be written as the ratio of two integers. Any fraction can be converted to a decimal by dividing the numerator by the denominator.
5-3 Adding and Subtracting Fractions
To add or subtract fractions with unlike denominators, you first rewrite the fractions with a common denominator. It is often practical to use the LCD because the resulting numerators have the least possible values, so are easiest to add. Also, the result often is already in simplest form.
5-4 Multiplying and Dividing Fractions
When subtracting integers, you make two changes to the expression. You add the opposite of the second integer. When you divide fraction, you also make two changes. You multiply by the reciprocal of the second fraction, or divisor. Point out that in both cases you are performing two actions that, in effect, offset each other so that the value of the expression remains unchanged.
5-5 Customary Units of Measures
Note the difference between the metric system (Lesson 3-6) and the customary system. To convert units within the metric system, you multiply or divide by powers of 10. To convert units within the customary system, you multiply or divide by 3, 8, 12, 16 or some other number, depending on the units or equivalency.
5-6 Working Backward
In some problem-solving situations, it may be helpful to work backward. These situations often involve a time of day or a known end result of a process. To solve these problems, begin with the final result and work backward to the beginning.
5-7 Solving Equations by Adding and Subtracting Fractions
You use inverse operations and the Addition and Subtraction Properties of Equality to solve equations involving subtraction or addition of fractions. You often must find the LCD of the fractions to complete the solution.
5-8 Solving Equations by Multiplying Fractions
When a variable in an equation is multiplied by a fraction, you divide each side of the equation by that fraction to solve the equation. Therefore, to solve an equation where the variable is multiplied by a fraction, you multiply each side of the equation by the reciprocal of that fraction.
5-9 Powers of Products and Quotients
When simplifying the power of a product, you can write the factors and then rearrange them using the Commutative and Associative Properties of Multiplication. The rule for raising a product to a power is similar to (but not the same as) the Distributive Property: the exponent outside the parenthesis is “distributed to” and multiplied by the exponents of each factor inside the parenthesis, including the exponent 1.
5-1 Comparing and Ordering Rational Numbers
A good method for comparing two fractions is to find a common denominator. The least common denominator (LCD) is the least common multiple (LCM) of the denominators. You can list the multiples of each number until you find a common multiple. Or, you can write the prime factorization of each number and use the greatest power of each prime factor. We use the phrase “take the left overs” after highlighting the common factors in the numerator and denominator. Listing the multiples may be easier for lesser numbers whereas prime factorization may be best for greater ones and numbers that include variables.
5-2 Fractions and Decimals
A rational number is any number you can write as a ratio a/b of two integers a and b, where b does not equal 0. Both terminating and repeating decimals can be written as the ratio of two integers. Decimals that are neither terminating nor repeating are irrational numbers and cannot be written as the ratio of two integers. Any fraction can be converted to a decimal by dividing the numerator by the denominator.
5-3 Adding and Subtracting Fractions
To add or subtract fractions with unlike denominators, you first rewrite the fractions with a common denominator. It is often practical to use the LCD because the resulting numerators have the least possible values, so are easiest to add. Also, the result often is already in simplest form.
5-4 Multiplying and Dividing Fractions
When subtracting integers, you make two changes to the expression. You add the opposite of the second integer. When you divide fraction, you also make two changes. You multiply by the reciprocal of the second fraction, or divisor. Point out that in both cases you are performing two actions that, in effect, offset each other so that the value of the expression remains unchanged.
5-5 Customary Units of Measures
Note the difference between the metric system (Lesson 3-6) and the customary system. To convert units within the metric system, you multiply or divide by powers of 10. To convert units within the customary system, you multiply or divide by 3, 8, 12, 16 or some other number, depending on the units or equivalency.
5-6 Working Backward
In some problem-solving situations, it may be helpful to work backward. These situations often involve a time of day or a known end result of a process. To solve these problems, begin with the final result and work backward to the beginning.
5-7 Solving Equations by Adding and Subtracting Fractions
You use inverse operations and the Addition and Subtraction Properties of Equality to solve equations involving subtraction or addition of fractions. You often must find the LCD of the fractions to complete the solution.
5-8 Solving Equations by Multiplying Fractions
When a variable in an equation is multiplied by a fraction, you divide each side of the equation by that fraction to solve the equation. Therefore, to solve an equation where the variable is multiplied by a fraction, you multiply each side of the equation by the reciprocal of that fraction.
5-9 Powers of Products and Quotients
When simplifying the power of a product, you can write the factors and then rearrange them using the Commutative and Associative Properties of Multiplication. The rule for raising a product to a power is similar to (but not the same as) the Distributive Property: the exponent outside the parenthesis is “distributed to” and multiplied by the exponents of each factor inside the parenthesis, including the exponent 1.